Million Dollar Mathematics!

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Million Dollar Mathematics!

Alissa S. Crans

Loyola Marymount University

Southern California Undergraduate Math Day University of California, San Diego

April 30, 2011

This image is from the Wikipedia article on Minesweeper.

Permission to use it was granted under the GNU Free Documentation License.

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Clay Mathematics Institute

• Cambridge, Massachusetts

• Founded by Landon T. Clay in 1998

Curiosity is part of human nature. Unfortunately, the established religions no longer provide the answers that

are satisfactory, and that translates into a need for

certainty and truth. And that is what makes mathematics work, makes people commit their lives to it. It is the

desire for truth and the response to the beauty and elegance of mathematics that drives mathematicians.

- Landon T. Clay

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Clay Mathematics Institute

The primary objectives and purposes:

• to increase and disseminate mathematical knowledge,

• to educate mathematicians and other scientists about new discoveries in the field of mathematics,

• to encourage gifted students to pursue mathematical careers,

• and to recognize extraordinary achievements and

advances in mathematical research.

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History of Mathematical Prizes

• 18th Century: Academies in Berlin, Paris, and St. Petersburg

Daniel Bernoulli 1700 - 1782

Leonhard Euler 1707 - 1783

Joseph-Louis Lagrange 1736 - 1813

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History of Mathematical Prizes

• 19th Century: Paris Academy, Grand Prix, Steiner Prize, Jablonowski Society, Danish Academy of

Sciences, Wolfskehl Prize

These images are from the Wikipedia articles on Sophie Germain, Augustin-Louis Cauchy, and Andrew Wiles.

The first and second are in the public domain.

Sophie Germain 1776 - 1831

Augustin-Louis Cauchy 1789 - 1857

Andrew Wiles 1953 -

copyright C. J. Mozzochi, Princeton N.J

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History of Mathematical Prizes

• Fields Medal, established 1930s

John Charles Fields 1863 - 1932

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Hilbert’s 23 Problems

Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the

coming developments of our science and at the secrets of its development in the centuries to come?

What will be the ends toward which the spirit of

future generations of mathematicians will tend? What methods, what new facts will the new century reveal

in the vast and rich field of mathematical thought?

This image is from the Wikipedia article on David Hilbert. It is in the public domain.

David Hilbert 1862 - 1943

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Hilbert’s 23 Problems

•

There is no set whose cardinality is strictly between that of the integers and that of the real numbers

(Continuum Hypothesis)

•

Given any two polyhedra of equal volume, is it always

possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

•

Find an algorithm to determine whether a given polynomial with integer coefficients has an integer solution.

•

Riemann Hypothesis

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The Millennium Prize Problems

• P vs NP

• Yang-Mills Theory

• Hodge Conjecture

• Riemann Hypothesis

• Poincaré Conjecture

• Navier-Stokes Equations

• Birch and Swinnerton-Dyer Conjecture

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The Millennium Prize Problems

• ^{P vs NP}

(computer science)

• Yang-Mills Theory

^(physics)

• Hodge Conjecture

(algebraic geometry)

• Riemann Hypothesis

(number theory)

• Poincaré Conjecture

^(topology)

• Navier-Stokes Equations

(differential equations)

• Birch and Swinnerton-Dyer Conjecture

(number theory)

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The Millennium Prize Problems

• P vs NP

• Yang-Mills Theory

• Hodge Conjecture

• Riemann Hypothesis

• Poincaré Conjecture

• Navier-Stokes Equations

• Birch and Swinnerton-Dyer Conjecture

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Poincaré Conjecture

A blob (satisfying certain features) in 3

dimensions is a sphere if every loop on it can

be deformed to a point.

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Poincaré Conjecture

This image is from the Wikipedia article on the torus.

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Poincaré Conjecture

A blob (satisfying certain features) in 3 dimensions is a

sphere if every loop on it can be deformed to a point.

Poincaré Conjecture:

The same is true for blobs in 4 dimensions.

Henri Poincaré 1854 - 1912

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• 1961 - Stephen Smale proved true in dimensions > 6

• 1982 - Michael Freedman proved true in dimension 5

• 2002 - Grigori Perelman proved the conjecture

Poincaré Conjecture

Grigori Perelman 1966 -

This image is from the Wikipedia article on Grigori Perelman.

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Poincaré Conjecture

Grigory Perelman, the maths genius who said no to $1m

Perelman cracks a century-old conundrum, refuses the reward, and barricades himself in his flat

(Guardian, UK)

A Math Problem Solver Declines a $1 Million Prize

(New York Times)

BREAKTHROUGH OF THE YEAR

The Poincaré Conjecture--Proved

(Science Magazine 2006)

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These images are from the Wikipedia articles on Stephen Cook and Leonid Levin.

Permission to use them was granted under the GNU Free Documentation License.

P vs. NP Problem

Can every problem whose solution can be efficiently checked by a computer also be

efficiently solved by a computer?

Stephen A. Cook 1939 -

Leonid Levin 1948 -

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P vs. NP Problem

Polynomial Time Process:

Given data of size N, it takes at most cN

^k

basic

steps to complete the task.

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P vs. NP Problem

Polynomial Time Process:

Given data of size N, it takes at most cN

^k

basic steps to complete the task.

Example: Product of two N-digit numbers 5927

x 6384

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P vs. NP Problem

Polynomial Time Process:

Given data of size N, it takes at most cN

^k

basic steps to complete the task.

Example: Product of two N-digit numbers 5927

x 6384

Approximately 2N²steps

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Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

P vs. NP Problem

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

16 steps

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

~16 more steps

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P vs. NP Problem

Our basic operations are multiplying two single digits or adding two single digits.

5927 x 6384

23708

47416 17781

35562

37,837,968

N = 4

~ 2(16) = 2N²steps

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P vs. NP Problem

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P vs. NP Problem

Example:

Suppose you are a salesman and you must visit

each of N cities. How many possible routes do

you have?

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P vs. NP Problem

Example:

Suppose you are a salesman and you must visit each of N cities. How many possible routes do you have?

N ^x

(N - 1)

^x

(N - 2)

^x

...

^x

3

^x

2

^x

1 = N!

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P vs. NP Problem

Example:

Suppose you are a salesman and you must visit each of N cities. How many possible routes do you have?

N ^x

(N - 1)

^x

(N - 2)

^x

...

^x

3

^x

2

^x

1 = N!

N! > 2^N^{(N > 4)}

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P vs. NP Problem

Example:

Suppose you are a salesman and you must visit each of N cities. How many possible routes do you have?

N ^x

(N - 1)

^x

(N - 2)

^x

...

^x

3

^x

2

^x

1 = N!

N! > 2^N^{(N > 4)}

Exponentially many steps!

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P vs. NP Problem

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How long? If N = 50, it takes a computer

P vs. NP Problem

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How long? If N = 50, it takes a computer .125 seconds to compute N

³

steps

P vs. NP Problem

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How long? If N = 50, it takes a computer

.125 seconds to compute N

³

steps 35 years to compute 2

^N

steps

P vs. NP Problem

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How long? If N = 50, it takes a computer

.125 seconds to compute N

³

steps 35 years to compute 2

^N

steps

200,000,000 centuries to compute 3

^N

steps

P vs. NP Problem

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P vs. NP Problem

Class P Problems:

The collection of yes/no problems that can be solved in polynomial time. (in cN

^k

steps)

quickly solvable

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P vs. NP Problem

Class P Problems:

The collection of yes/no problems that can be solved in polynomial time. (in cN

^k

steps)

quickly solvable

Examples:

• Given a positive integer, is it prime?

• Given a graph, does it have an Euler cycle?

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P vs. NP Problem

An Euler cycle is a route in a graph that starts and ends at the same vertex and traverses every edge once.

Example:

^a

b

c

e d

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P vs. NP Problem

An Euler cycle is a route in a graph that starts and ends at the same vertex and traverses every edge once.

Example:

^a

b

c

e d

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P vs. NP Problem

An Euler cycle is a route in a graph that starts and ends at the same vertex and traverses every edge once.

Example:

^a

b

c

e d

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P vs. NP Problem

An Euler cycle is a route in a graph that starts and ends at the same vertex and traverses every edge once.

Example:

^a

b

c

e d

Using brute force, it takes n! basic steps to determine whether a graph with n edges has an Euler cycle! (not polynomial time!)

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P vs. NP Problem

Euler proved that a graph has an Euler cycle if and only if every vertex has an even number of edges coming into it.

Example:

^a

b

c

e d

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P vs. NP Problem

Euler proved that a graph has an Euler cycle if and only if every vertex has an even number of edges coming into it.

That takes polynomial time to check!

Example:

^a

b

c

e d

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P vs. NP Problem

Class NP Problems:

The collection of yes/no problems whose

solutions can be verified in polynomial time.

(in cN

^k

steps)

quickly checkable, but not quickly solvable

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P vs. NP Problem

Class NP Problems:

The collection of yes/no problems whose

solutions can be verified in polynomial time.

(in cN

^k

steps)

quickly checkable, but not quickly solvable

Examples of quickly checkable:

• Dorm assignments

• Jigsaw puzzles

• Minesweeper

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P vs. NP Problem

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P vs. NP Problem

Examples:

• Given a set of integers, does there exist a

nonempty subset whose elements sum to zero?

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P vs. NP Problem

Examples:

• Given a set of integers, does there exist a

nonempty subset whose elements sum to zero?

{-2, -3, 14, 7, 15, -10}

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P vs. NP Problem

Examples:

• Given a set of integers, does there exist a

nonempty subset whose elements sum to zero?

{-2, -3, 14, 7, 15, -10}

yes! {-2, -3, -10, 15}

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P vs. NP Problem

Examples:

• Given a set of integers, does there exist a

nonempty subset whose elements sum to zero?

{-2, -3, 14, 7, 15, -10}

yes! {-2, -3, -10, 15}

• Given a graph with n vertices, can we color the vertices with k colors so that no two adjacent

vertices have the same color?

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P vs. NP Problem

Examples:

• Given a graph, does it have a Hamiltonian cycle?

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P vs. NP Problem

A Hamiltonian cycle is a route in a graph that starts and ends at the same vertex and traverses every vertex exactly once.

Examples:

• Given a graph, does it have a Hamiltonian cycle?

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P vs. NP Problem

A Hamiltonian cycle is a route in a graph that starts and ends at the same vertex and traverses every vertex exactly once.

Example:

^a

b

c

e d

Examples:

• Given a graph, does it have a Hamiltonian cycle?

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P vs. NP Problem

A Hamiltonian cycle is a route in a graph that starts and ends at the same vertex and traverses every vertex exactly once.

Example:

^a

b

c e

No

Hamiltonian cycle

Examples:

• Given a graph, does it have a Hamiltonian cycle?

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P vs. NP Problem

The Million Dollar Problem:

Is the collection of P problems equal to the

collection of NP problems?

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P vs. NP Problem

The Million Dollar Problem:

Is the collection of P problems equal to the collection of NP problems?

Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be

computed quickly?

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P vs. NP Problem

The Million Dollar Problem:

Is the collection of P problems equal to the collection of NP problems?

Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be

computed quickly?

Is it harder to solve a problem yourself than to check that a solution by someone else is correct?

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P vs. NP Problem

Solution?

Most mathematicians and computer scientists

believe that P = NP.

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P vs. NP Problem

Solution?

Most mathematicians and computer scientists believe that P = NP.

If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in "creative leaps," no fundamental

gap between solving a problem and recognizing the solution once it's found.

Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss...

- Scott Aaronson, MIT

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P vs. NP Problem

Solution?

Most mathematicians and computer scientists believe that P = NP.

If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in "creative leaps," no fundamental

gap between solving a problem and recognizing the solution once it's found.

Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss...

- Scott Aaronson, MIT

The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument.

- Moshe Y. Vardi, Rice University

Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem.

- Anil Nerode, Cornell University

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P vs. NP Problem

Status?

In August 2010, Vinay Deolalikar of HP Labs offered a proposed proof that P = NP. It was found to be flawed.

Step 1: Post Elusive Proof. Step 2: Watch Fireworks.

(New York Times)

Million dollar maths puzzle sparks row

(BBC News)